cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A263968 a(n) = Li_{-n}(phi) + Li_{-n}(1-phi), where Li_n(x) is the polylogarithm, phi=(1+sqrt(5))/2 is the golden ratio.

Original entry on oeis.org

-3, 4, -18, 112, -930, 9664, -120498, 1752832, -29140290, 545004544, -11325668178, 258892951552, -6456024679650, 174410345857024, -5074158021135858, 158168121299894272, -5258993667674555010, 185786981314092335104, -6949466928081909755538
Offset: 0

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Author

Vladimir Reshetnikov, Oct 30 2015

Keywords

Comments

2*Li_{-n}(phi) = a(n) - (-1)^n*A000557(n)*sqrt(5), so a(n) represents integer terms in 2*Li_{-n}(phi), and A000557(n) (with alternating signs) represents terms proportional to sqrt(5).

Examples

			For n = 4, Li_{-4}(phi) = -930 - 416*sqrt(5), so a(4) = -930 and A000557(4) = 416.

Links

Crossrefs

Cf. A000032, A000556, A000557, A048993.

Programs

  • Maple
    a := n -> polylog(-n,(1+sqrt(5))/2)+polylog(-n,(1-sqrt(5))/2):
seq(round(evalf(a(n),32)), n=0..18); # Peter Luschny, Nov 01 2015
  • Mathematica
    Round@Table[PolyLog[-n, GoldenRatio] + PolyLog[-n, 1-GoldenRatio], {n, 0, 20}]
Table[(-1)^(n+1) Sum[k! LucasL[k+2] StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
  • PARI
    vector(100, n, n--; (-1)^(n+1)*sum(k=0, n, k!*stirling(n, k, 2)*(2*fibonacci(k+1) + fibonacci(k+2)))) \\ Altug Alkan, Oct 31 2015

Formula

a(n) = (-1)^(n+1)*Sum_{k=0..n} k!*Lucas(k+2)*Stirling2(n,k), where Lucas(n) = A000032(n) and A048993(n,k) = Stirling2(n,k).
a(n) = (-1)^(n+1)*(2*A000556(n) + A000557(n)).
E.g.f.: -(1+2*exp(x))/(1+2*sinh(x)).
a(n) ~ (-1)^(n+1) * n! / log((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, Oct 31 2015

A332257 E.g.f.: (1 - sinh(x)) / (1 - 2*sinh(x)).

Original entry on oeis.org

1, 1, 4, 25, 208, 2161, 26944, 391945, 6515968, 121866721, 2532496384, 57890223865, 1443611004928, 38999338931281, 1134616226381824, 35367467110007785, 1175946733416153088, 41543231955279099841, 1553948045857778827264, 61355543097139813855705
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2020

Keywords

Links

Crossrefs

Cf. A000557, A000670, A002866, A006154, A050351, A331608.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[(1 - Sinh[x])/(1 - 2 Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    seq(n)={Vec(serlaplace((1 - sinh(x + O(x*x^n))) / (1 - 2*sinh(x + O(x*x^n)))))} \\ Andrew Howroyd, Feb 08 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A006154(k) * a(n-k).
a(n) ~ n! / (2*sqrt(5) * log((1 + sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, Feb 08 2020

A358031 Expansion of e.g.f. (1 - log(1-x))/(1 + log(1-x) * (1 - log(1-x))).

Original entry on oeis.org

1, 2, 8, 52, 450, 4878, 63474, 963744, 16724016, 326497632, 7082393136, 168995017200, 4399028766192, 124051494462816, 3767315220903072, 122581568808533760, 4254486275273419008, 156890997080103149568, 6125936704495619486976, 252480641031903073955328
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2022

Keywords

Crossrefs

Cf. A354013, A354018.
Cf. A000557, A358032.

Programs

  • Maple
    f:= proc(n) local k; add(k!*combinat:-fibonacci(k+2)*abs(Stirling1(n,k)),k=0..n) end proc:
map(f, [$0..30]); # Robert Israel, Oct 25 2022
  • Mathematica
    With[{nn=20},CoefficientList[Series[(1-Log[1-x])/(1+Log[1-x](1-Log[1-x])),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 25 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1-log(1-x))/(1+log(1-x)*(1-log(1-x)))))
  • PARI
    a(n) = sum(k=0, n, k!*fibonacci(k+2)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} k! * Fibonacci(k+2) * |Stirling1(n,k)|.
a(n) = A354013(n) + A354018(n).

A358032 Expansion of e.g.f. (1 + log(1+x))/(1 - log(1+x) * (1 + log(1+x))).

Original entry on oeis.org

1, 2, 4, 16, 66, 438, 2694, 25296, 204576, 2509728, 24912816, 381010320, 4440815472, 82150191264, 1089159690912, 23879423005440, 351430312958208, 9005004020293632, 144184020764472576, 4277182103330660352, 73227226213747521792, 2499666592623881921280
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2022

Keywords

Crossrefs

Cf. A005444, A005445.
Cf. A000557, A358031.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1+log(1+x))/(1-log(1+x)*(1+log(1+x)))))
  • PARI
    a(n) = sum(k=0, n, k!*fibonacci(k+2)*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} k! * Fibonacci(k+2) * Stirling1(n,k).
a(n) = A005444(n) + A005445(n).

A364822 Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).

Original entry on oeis.org

1, 2, 9, 56, 465, 4832, 60249, 876416, 14570145, 272502272, 5662834089, 129446475776, 3228012339825, 87205172928512, 2537079010567929, 79084060649947136, 2629496833837277505, 92893490657046167552, 3474733464040954877769, 137195165161622584426496, 5702069567580948171751185
Offset: 0

Views

Author

Mélika Tebni, Nov 07 2023

Keywords

Comments

Conjectures: For p prime (p > 2), a(p) == 2 (mod p).
For n = 2^m (m natural number), a(n) == 1 (mod n).

Links

Crossrefs

Cf. A000556, A000557, A005923, A332257, A341724.

Programs

  • Maple
    a := n -> add(binomial(n,2*k)*add(j!*combinat[fibonacci](j+2)*Stirling2(n-2*k,j), j=0..n-2*k), k=0..floor(n/2)):
seq(a(n), n = 0 .. 20);
# second program:
b := proc(n) option remember; `if`(n = 0, 1, 2+2*add(binomial(n,2*k-1)*b(n-2*k+1), k=1..floor((n+1)/2))) end:
a := proc(n) `if`(n = 0, 1, b(n)/2) end: seq(a(n), n = 0 .. 20);
# third program:
(1/2)*((exp(-x) + exp(x))/(1 + exp(-x) - exp(x))): series(%, x, 21):
seq(n!*coeff(%, x, n), n = 0..20);  # Peter Luschny, Nov 07 2023
  • Mathematica
    a[n_]:=n!*SeriesCoefficient[Cosh[x]/(1 - 2*Sinh[x]),{x,0,n}]; Array[a,21,0] (* Stefano Spezia, Nov 07 2023 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(cosh(x) / (1 - 2*sinh(x)))) \\ Michel Marcus, Nov 07 2023

Formula

a(n) = A000556(n) + A332257(n), for n > 0.
a(n) = (-1)^n*Sum_{k=0..floor(n/2)} A341724(n,2*k).
a(n) = (A000556(n) + A005923(n)) / 2.
a(n) ~ n! / (2*log((1 + sqrt(5))/2)^(n+1)).

A005924 From solution to a difference equation.

Original entry on oeis.org

1, 7, 49, 415, 4321, 53887, 783889, 13031935
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Formula

Numbers so far satisfy a(n) = A000557(n) - 1. - Ralf Stephan, May 23 2004

A107404 Expansion of e.g.f. 1/(1 - sinh(x))^2.

Original entry on oeis.org

1, 2, 6, 26, 144, 962, 7536, 67706, 685824, 7730882, 95970816, 1300815386, 19113775104, 302616787202, 5135568746496, 92996021795066, 1789758460329984, 36479831022049922, 785020114093080576, 17785273588395966746, 423150055005134782464, 10548427254444904799042
Offset: 0

Views

Author

Miklos Kristof, Jun 09 2005

Keywords

Crossrefs

Cf. A000557. A006154, A136630.

Programs

  • Maple
    E(x):=1/(1-sinh(x))^2: f[0]:=E(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..30);
  • Mathematica
    CoefficientList[Series[1/(1-Sinh[x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (k+1)!*a136630(n, k)); \\ Seiichi Manyama, Feb 17 2025

Formula

a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A006154. - Peter Bala, Dec 06 2011
a(n) ~ n!*n/(2*(log(1+sqrt(2)))^(n+2)). - Vaclav Kotesovec, Jun 27 2013
a(n) = Sum_{k=0..n} (k+1)! * A136630(n,k). - Seiichi Manyama, Feb 17 2025

A367887 Expansion of e.g.f. exp(2*x) / (1 - 2*sinh(x)).

Original entry on oeis.org

1, 4, 20, 130, 1088, 11314, 141080, 2052250, 34118048, 638102434, 13260323240, 303117147370, 7558845354608, 204203189722354, 5940927689713400, 185186461979970490, 6157337034085736768, 217523186522883467074, 8136577601614291359560, 321261794453042025993610, 13352198666907246870560528
Offset: 0

Views

Author

Mélika Tebni, Dec 04 2023

Keywords

Links

Crossrefs

Cf. A000045, A000557, A005923, A341723, A341724, A341725.

Programs

  • Maple
    a := n -> -1-0^n+add(k!*combinat[fibonacci](k+4)*Stirling2(n, k), k = 0 .. n):
seq(a(n), n=0..20);
# second program:
a := proc(n) option remember; `if`(n=0,1,3^n+add((2^(n-k)-1)*binomial(n, k)*a(k), k=0..n-1)) end:
seq(a(n), n=0..20);
# third program:
a := n -> add(2^k*binomial(n, k)*add(j!*combinat[fibonacci](j+2)*Stirling2(n-k, j), j=0..n-k), k=0..n):
seq(a(n), n=0..20);
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(2*x) / (1 - 2*sinh(x)))) \\ Michel Marcus, Dec 04 2023

Formula

a(n) = Sum_{k=0..n} A341725(n,k).
a(n) = (-1)^n*Sum_{k=0..n} (-2)^k*A341724(n,k).
a(n) = -1-0^n+Sum_{k=0..n} k!*Fibonacci(k+4)*Stirling2(n,k).
a(0) = 1; a(n) = 3^n+Sum_{k=0..n-1} (2^(n-k)-1)*binomial(n,k)*a(k).
a(n) ~ n! * (phi)^2 / (sqrt(5) * (log(phi))^(n+1)), where phi is the golden ratio.
a(n) = -1 + A000557(n) + A005923(n) = -1 + Sum_{k=0..n} |A341723(n,k) + A341724(n,k)|.
Previous Showing 11-18 of 18 results.

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